Hi! Well, Fort Myers is a bit of a difficult place to get around without a car, in my opinion. I guess it depends on what you would like to do while you are here.
If you want to be close be within walking distance of a beach, you can either stay on Fort Myers Beach (it is an island) or Sanibel/Captiva islands. Those are the only places with hotels and beaches, as far as I know. Fort Myers Beach is certainly a little more touristy than Sanibel, I would do some research on which place would be more your style. Sanibel is pretty quiet and "natural," and FMB is more "spring break."
Both places would be appropriate for having things in walking distance, perhaps FMB more so than Sanibel. But if you wanted to do anything in Fort Myers proper, you'd really need a car. Public transportation is mostly non-existent. How long would you be staying for?
As far as hotels- any hotel on Fort Myers Beach would be close to the beach, the island is very narrow. Even if you stay somewhere on the bay side of the island, the beach is just a short walk across the street. The northwest end of Fort Myers Beach is where all of the "stuff" is: restaurants, hotels, shops. The further south you go on the island, the less stuff. Anything with an address higher than 3000 on Estero Blvd. is probably too far away if you want to be close to things. The only thing I can say about specific hotels is that I would stay away from the Lani Kai. It is a spring break hotel, and the facilities look less than well maintained. I've never heard anyone say anything nice about the rooms there. I hear good things about the Pink Shell, although I think it is a little pricey.
Sanibel is a little bigger and more spread out. I don't know too much about any of the hotels out there.
If you want to stay in Fort Myers proper and go to a beach, you'll want to rent a car. There aren't a lot of areas in Fort Myers than could be described as good for walking around in.
Hope some of this helps, I'd be glad to give you more specifics if you want.